The one and the multiple in Badiou’s ontology

Hegel argued that a philosophical system could not, in the final analysis, be conceived entirely separately from the method by which it was presented. If this judgement still holds today, it must surely apply to Badiou, one of the few contemporary philosophers to declare that systematic ontology is both possible and necessary, and whose major work Being & Event is devoted to laying out such a systematic ontology.

So we can be sure that Badiou’s ontology is oriented by the fact that its opening meditation sets out from the question of the one versus the multiple. The question – explored in this short essay – is over the nature of this orientation. I argue that the decision affects Badiou’s system on two distinct levels: there are effects arising from the specific manner in which he privileges the multiple over the one, and there are effects arising from the very fact he starts with the problem of multiplicity.

The first of these levels is more superficial than the second – in other words, one can concoct mathematicised ontologies that broadly resemble Badiou’s but which diverge on specific questions such as the existence of the Whole or the status of Atoms. However, it is harder to see how Badiou could have produced a set theory based ontology that did not have to resolve the metaphysical problems surrounding multiplicity fairly early on. If Badiou had started from a different question, his system would look and feel different.

Before getting into the detail of Badiou’s ontology as laid out in B&E, I’ll start with a brief look at the some of the context surrounding the question of multiplicity. The first thing to note is that Badiou’s decision to start from this issue was by no means unusual or unprecedented in late 20th century French philosophy. The rise of structuralism in the 1960s inevitably raised questions over the ontological status of its systems of signs.

These questions became more pressing as it became clear that individual signs were being almost entirely absorbed into those systems, their positive identity replaced by a collection of differential relationships with other signs. Derrida embarked on his critique of the metaphysics of identity, while Deleuze put forward his own concept of pure multiplicity. The notion of a multiple conceptualised without any prior reference to a one was already well established in France by the time Badiou wrote B&E in the mid-1980s.

The question of the multiple had a political and scientific context too. Liberal ideology in all its variations tended to privilege sovereign individuals over groups or classes. Socialists tended to make the opposite choice, emphasising collective subjectivity over individual rationality. And meanwhile mathematicians had developed and honed their own theory of multiplicity – set theory – in response to paradoxes they had encountered during the previous few decades.

Badiou’s distinctive approach to the problem of multiplicity arose, roughly speaking, from combining these three currents. But it was his use of set theory that marked him off both from other contemporary approaches to the issue and from those found in the philosophical tradition. That tradition stretches back a fair way, as Badiou makes clear in the opening paragraph of B&E’s first meditation.

He quotes a formulation from Leibniz that seemingly chains being to the one and downgrades the multiple: “Ce qui n’est pas un être n’est pas un être.” (“What is not a being is not a being.”) He mentions Plato’s Parmenides dialogue, whose latter half is devoted to a bewildering series of paradoxical arguments over the ontological rivalry between ones and multiples (a dialogue examined in more detail in the second meditation).

Badiou even offers his own summary of the metaphysical chicken-and-egg problem at hand: “What presents itself is essentially multiple; what presents itself is essentially one.” A thing might initially seem to present itself as a unitary one, but in the very act of presenting itself it unravels, so to speak, into a multiple: my dinner resolves into {fish, chips, peas} for example.

Having set out the problem, Badiou proceeds to make a sudden and controversial move. He cuts through the ontological Gordian knot by announcing a decision: the one is not. From now on he is going to privilege the multiple over the one in terms of ontological priority, and proceed to systematically examine the consequences of this decision for ontology.

Badiou rapidly supplements his declaration with a slightly more mysterious one: “il y a de l’Un” (“there is Oneness”). We will examine what this means shortly. But before that there are two important points to note about the decision that the one is not.

The first is that Badiou is not suggesting that everyone who came before him was simply wrong on this question, or stupid. His position is rather that the advent of modern mathematics – and set theory in particular – makes it possible for the first time to develop a systematic ontology based around the multiple rather than the one.

This is set out more clearly in the introduction: “The science of being qua being has existed since the Greeks – such is the sense and status of mathematics. However, it is only today we have the means to know this.” (B&E, p3) In Badiou’s terminology, there has been a scientific event, organised around the names Cantor-Gödel-Cohen, that has fundamentally transformed the way we understand the ontological situation. Following the consequences of this new insight is part of Badiou’s subjective fidelity to that event.

The second point is that even within the broad parameters of a set theory based ontology, Badiou could have made a different decision. To understand this we need to think a little bit more about how sets are structured.

In the third meditation of B&E, Badiou notes how the paradoxes of naïve set theory forced mathematicians to adopt an axiomatic approach to set theory based around undefined abstract signs: “It is necessary to abandon all hope of explicitly defining the notion of a set.

Neither intuition nor language are capable of supporting the pure multiple – such as founded by the sole relation ‘belonging to’, written ∈ – being counted-as-one in a universal concept.” (B&E, p43). The ∈ symbol is the fundamental undefined relation of set theory. It is asymmetric, in that x ∈ y does not imply that y ∈ x. In fact in a phrase like x ∈ y, the multiple x is usually thought of as an ‘element’ and the multiple y is thought of as a ‘collection’. But these determinations are relative to the statement x ∈ y, they say nothing about the nature of x and y themselves.

One of the key features of Zermelo-Fraenkel (ZF) set theory – the axiomatisation of sets favoured by Badiou and by most mathematicians – is that every entity is a pure multiple, ie a set that can in principle appear on either side of the ∈ sign. But there are alternative axiomatisations where this is not the case. For instance, we could allow certain entities to appear on the left of ∈ but not the right. This would correspond to things that could be members of sets but not have any members themselves – what mathematicians call atoms.

Conversely, we could allow entities that can appear on the right but not the left. These would be so-called classes – collections of sets (or atoms) that could not, however, themselves be members of sets or classes. In a sense both atoms and classes represent deviations from the notion of a pure consistent multiple, metaphysically speaking. An atom corresponds to the “little one” you get when you examine the elements that comprise a multiple. A class corresponds to the “inconsistent multiple” you get when a multiple is considered without the implicit gathering-together that makes a multiple of it.

The variations on ZF that include atoms or classes (or both) are perfectly legitimate from a mathematical point of view. One can in fact prove that if ZF is consistent, so are they (and vice versa). So Badiou could have chosen a variant ontology that included atoms or classes. But this would not be an ontology based on pure multiples of multiples. It would be imply that a different path had been taken to resolve the question of the one versus the multiple.

Such a course of action would have two drawbacks from a philosophical point of view, however. For starters it would not accord so well with mathematical practice among set theorists, who tend to treat ZF as their ‘plain vanilla’ axiomatisation. More seriously, perhaps, such variant systems would break Occam’s Razor by “needlessly” introducing different types of entities. They would typically involve a much messier and larger collection of axioms (the axiom of extensionality can’t easily distinguish between atoms and the void, for instance, since both have no members). And this in turn would contradict Aristotle’s dictum, quoted approvingly by Badiou on p60 of B&E, that the first principles should be “are as few as they are crucial”.

Returning to Badiou’s decision to grant sole ontological credentials to the multiple – what does this mean for the one? Does it simply disappear entirely? Not quite. We have already seen in our discussion of atoms and classes above that pure multiples can take on the ‘role’ of ones. For instance, the multiple x ‘acts’ as a one in the statement x ∈ y – it becomes an element of y, as Leibniz might put it.

It is this ‘virtual’ one that Badiou is referring to when he says “il y a de l’Un”. It does not exist as such (that privilege belongs solely to the multiple-that-is-presented), but is rather an ‘operational effect’ of the count-as-one that invariably accompanies presentation. This count-as-one underpins a subtle but important distinction between consistent and inconsistent multiplicity.

The multiples that are presented to us are consistent (in the etymological sense: they con-sist, stand together). But given a presented consistent multiple, one can retroactively posit what the multiple would have been prior to being counted-as-one, prior to becoming con-sistent. This is the inconsistent multiplicity that is never directly presented but can always be postulated as what-must-have-been before the count-as-one.

To return to an earlier example, I can imagine what the inconsistent multiple that became {fish, chips, peas} would look like – I just mentally ‘take the brackets off’. Note that the paradoxes of naïve set theory tell us that not every multiple can be counted as one. For instance, the so-called Russell set that comprises every set that is not an element of itself cannot be made to con-sist – it is a fundamentally inconsistent multiple. Different axiomatisations of set theory will involve different multiples becoming unpresentable in this manner.

Finally, one can apply this distinction between consistent and inconsistent multiplicity to the multiple-of-nothing, ie the void. The consistent multiple is usually written ∅ and corresponds to what Badiou sometimes calls ‘the name of the void’. The inconsistent multiple is the void proper and is harder to visualise – one can perhaps consider { } with the brackets taken off…

We have seen how certain variants of ZF set theory – ones involving atoms or classes – introduce changes to Badiou’s ontology that correspond to slightly different ontological decisions concerning the one and the multiple. These changes are, however, relatively superficial, in that any ontological theorem developed in one system can easily be adapted to the others.

The question remains as to whether or not Badiou’s focus on the question of the one and the multiple determines or shapes his system at a deeper level, or whether one can produce significantly different ontologies that nevertheless start out from that question. One approach would be to consider axioms that correspond to a very different type of set theory to ZF, rather than a minor variant.

One such system is New Foundations (NF), proposed by WVO Quine in 1937. NF has several unusual features that seem positively pathological to anyone used to ZF. You can prove that the Whole, the set of all sets, exists. The axioms of foundation and choice are provably false in NF, while the axiom of infinity can be proved without recourse to a new axiom. One could surmise that an NF-based ontology would correspond to a fully theological and idealist universe, where Spinoza’s God as Totality actually exists!

A cousin of NF known as NFU provides a natural setting for ‘mereology’ – the metaphysics of wholes and parts. An NFU-based ontology would arguably preserve Badiou’s basic ideas but take whole/parts as its fundamental metaphysical opposition rather than one/multiple (see Parts of Classes by David Lewis for more on this).

Nevertheless, both these ontologies would still be based on a version set theory, so one suspects that the aporias of the one and the multiple that have been around since Plato would soon resurface, albeit in altered form. The more intriguing question is whether one can remain faithful to Badiou’s “mathematics = ontology” injunction but not start from set theory (which is, after all, only a small region of mathematics and a relative backwater at that).

Badiou has himself done this to a certain extent in Logics of Worlds, which deploys category theory rather than set theory. But Badiou insists that this ‘logic’ supplements his set-based fundamental ontology rather than replaces it. Other mathematicians are more radical.

FW Lawvere, for instance, has spent decades arguing that categories suffice as a foundation for mathematics and that set theory can be safely junked (or rather rewritten on categorical lines). If such projects are feasible – and they are still the subject of fierce debate among mathematicians – it opens the prospect of a radically different mathematical ontology that corresponds more to what he calls the ‘constructivist’ orientation in thought.

One suspects Badiou would not be best pleased by such a development should it transpire. Ultimately this speculation underlines an important philosophical point: for all his advocacy of particular ontological decisions, Badiou’s system is in the final analysis a “meta-ontological” one that is open to transformation by mathematical events. The ontology of Being & Event corresponds to the maths of 1960s, that of Logics of Worlds to that of the 1980s. Future scientific revolutions will impact on it further – and in ways that by definition cannot be foreseen in advance.